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Rating: 
Summary: ODE
Review: Much material is included in this text, which is the first volume in a set of three and is suitable for graduate students and professionals. Chapter one details much material about ODE focusing on vector fields and much more including: Lie brackets, differential forms, geodesics, variable coefficient linear ODE, and of course, existence and uniqueness. Subsequent chapters cover Laplace's equation, Fourier Analysis and its relations to constant - coefficient linear PDE, linear Elliptic equations, Sobolev spaces, and linear evolution equations. Finally two important appendices are indcluded: One concerning functional analysis and another concerning differential geometry.While these chapters are essential self contained the chapter on sobolev spaces is important for later volumes. Overall the writing is rigorous and demanding --- as most mathemtical texts of this level are. However, with the assistance of a good library and dilligent effort *much* can be obtained from this set of texts, which explains precisely what is currently known about the genernal theory of PDE.
Rating:
Summary: ODE
Review: Much material is included in this text, which is the first volume in a set of three and is suitable for graduate students and professionals. Chapter one details much material about ODE focusing on vector fields and much more including: Lie brackets, differential forms, geodesics, variable coefficient linear ODE, and of course, existence and uniqueness. Subsequent chapters cover Laplace's equation, Fourier Analysis and its relations to constant - coefficient linear PDE, linear Elliptic equations, Sobolev spaces, and linear evolution equations. Finally two important appendices are indcluded: One concerning functional analysis and another concerning differential geometry. While these chapters are essential self contained the chapter on sobolev spaces is important for later volumes. Overall the writing is rigorous and demanding --- as most mathemtical texts of this level are. However, with the assistance of a good library and dilligent effort *much* can be obtained from this set of texts, which explains precisely what is currently known about the genernal theory of PDE.
Rating: 
Summary: Complete, accurate and well-written.
Review: The author accomplished the goal of presenting this broad and many-faceted subject in a thorough and comprehensive manner. Beginning from the fundamentals of ODE theory to the most sophisticated methods for solving important PDE's of mathematical physics, this series of three volumes comprises all what a modern analyst must know about the topic and much more. The contents of volume 1 are: Basic theory of ODE; Laplace and wave equations; Fourier analysis, distributions, and constant-coefficient linear PDE; Sobolev spaces; linear elliptic equations; linear evolution equations. Appendices: Outline of functional analysis; manifolds, vector bundles, and Lie groups. Originally intended for graduate students and working mathematicians, most of the material is suitable for advanced undergraduate courses. Includes excercises for each section and extensive references.
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